Single Event Probability (8.1.3) | CIE IGCSE Maths

Probability is a way to quantify the likelihood of events, ranging from events that cannot happen to those that are certain to happen. Understanding how to calculate the probability of a single event is crucial for making predictions based on a set of possible outcomes.

The Basic Principle

Probability is calculated as the ratio of the number of successful outcomes to the total number of possible outcomes. This can be expressed mathematically as:

P(A) = \frac{\text{Number of successful outcomes}}{\text{Total number of possible outcomes}}

Understanding Outcomes

Successful outcome: An outcome that fulfills the criteria of the event being considered.
Possible outcome: Any outcome that can possibly occur, regardless of whether it meets the criteria of the event.

Worked Examples

Example 1: Drawing an Ace from a Deck

Consider drawing an ace from a 52-card deck.

Number of successful outcomes (aces): 4
Total number of possible outcomes (cards in the deck): 52

P(\text{Ace}) = \frac{4}{52} = \frac{1}{13} \approx 0.0769

This calculation shows a $\approx 7.69\%$ chance of drawing an ace.

Example 2: Rolling a Die for a Number Greater Than 4

What is the probability of rolling a number greater than 4 on a standard six-sided die?

Number of successful outcomes: 2 (The numbers greater than 4 are 5 and 6.)
Total number of possible outcomes: 6 (A die has six faces.)

P(\text{Number} &gt; 4) = \frac{2}{6} = \frac{1}{3}

So, there is a 1 in 3 chance of rolling a number greater than 4.

Example 3: Coin Toss for Heads

For flipping a fair coin:

Successful outcomes (heads): 1
Possible outcomes (sides of the coin): 2

P(\text{Heads}) = \frac{1}{2} = 0.5

This means there's a 50% chance of flipping heads.

Practice Problems

Problem 1: Rolling a Sum of 7 with Two Dice

Calculate the probability of rolling a sum of 7 using two six-sided dice.

Successful outcomes (sum = 7): 6 $(1+6, 2+5, 3+4, 4+3, 5+2, 6+1)$
Total outcomes (all combinations of two dice): $6 \times 6 = 36$

P(\text{Sum} = 7) = \frac{6}{36} = \frac{1}{6} \approx 0.1667

This demonstrates a $\approx 16.67\%$ probability of rolling a sum of 7.

Problem 2: Picking a Red Ball

With a bag containing 5 red and 3 green balls, find the probability of drawing a red ball.

Successful outcomes (red balls): 5
Total outcomes (all balls): 8

P(Red) = \frac{5}{8} = 0.625

There's a 62.5% chance of drawing a red ball from the bag.

Written by:

Dr Rahil Sachak-Patwa

Oxford University - PhD Mathematics

Rahil spent ten years working as private tutor, teaching students for GCSEs, A-Levels, and university admissions. During his PhD he published papers on modelling infectious disease epidemics and was a tutor to undergraduate and masters students for mathematics courses.